There is a related notion of a \(k\) -collapsible simplicial complex. A simplicial complex \(X\)  is \(k\) -Leray (over a field \(\mathbb {k}\) , mostly chosen to be rationals), if the homology of every induced subcomplex of \(X\)  (a subcomplex of the form \(X[S]:=\lbrace A\in X\colon A\subseteq S\rbrace \)  with \(S\subseteq V(X)\) ) vanishes in dimension \(k\)  and larger (with coefficients in \(\mathbb {k}\) ). The Leray number \(\mathcal {L}(X)\)  of \(X\)  is the least integer \(k\)  for which \(X\)  is \(k\) -Leray. This coincides with the (Castelnuovo-Mumford) regularity of the Stanley-Reisner ring of \(X\)  (see [1]} for details). Wegner proved that the inequality \(\mathcal {L}(X)\le \mathcal {C}(X)\)  always holds [2]}, while
Matoušek and Tancer [3]} show that the gap \(\mathcal {C}(X)-\mathcal {L}(X)\)  between the collapsibility and Leray numbers could be arbitrarily large. On the other hand, we verify that they are equal when the simplicial complex \(X\)  is vertex decomposable.
